# Amortized quadrature for posterior expectations in inverse problems

arXiv: https://arxiv.org/abs/2606.15871 · HTML: https://mlsanity.com/paper/2606.15871/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 42 references that are in the map.
- First posted on arXiv 14 Jun 2026.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Kernel Thinning](https://mlsanity.com/paper/2105.05842/) (2021)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [Support points](https://mlsanity.com/paper/1609.01811/) (2016)
- [Conditional score-based diffusion models for Bayesian inference in infinite dimensions](https://mlsanity.com/paper/2305.19147/) (2023)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [Near-Optimal Coresets of Kernel Density Estimates](https://mlsanity.com/paper/1802.01751/) (2018)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Optimal Thinning of MCMC Output](https://mlsanity.com/paper/2005.03952/) (2020)
- [Positively Weighted Kernel Quadrature via Subsampling](https://mlsanity.com/paper/2107.09597/) (2021)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Distribution Compression in Near-linear Time](https://mlsanity.com/paper/2111.07941/) (2021)
- [Bayesian Quadrature for Multiple Related Integrals](https://mlsanity.com/paper/1801.04153/) (2018)
- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)
- [Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling](https://mlsanity.com/paper/2311.13548/) (2023)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [Amortized Monte Carlo Integration](https://mlsanity.com/paper/1907.08082/) (2019)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)
- [To discretize continually: Mean shift interacting particle systems for Bayesian inference](https://mlsanity.com/paper/2605.14142/) (2026)

## Abstract (arXiv)

Uncertainty in the solution of an inverse problem and in the tasks performed on it is quantified by posterior expectations, each an average of an integrand over M posterior samples. While designed quadratures improve on the O(M-1/2) error of Monte-Carlo estimation, they solve an optimization problem, often against the posterior density, for every new observation, which can be computationally costly. To address this limitation, we introduce the quadrature field, a set-equivariant network that maps an observation and its M posterior samples to an M-node signed-weight quadrature in one forward pass. Trained once on a family of posteriors to minimize the worst-case integration error over a class of functions, it serves any observation, any M and any integrand in that class with no further optimization. We show that, with high probability and up to a computable slack, the resulting quadrature is never worse than the Monte-Carlo estimate built from the same samples. We validate the quadrature field on closed-form and on learned posteriors, one constrained by a partial differential equation, where it improves on the Monte-Carlo estimate in median at every node count, often by orders of magnitude.